Charged particle
Moving charge = current
Associated magnetic field - B
Macroscopic picture
(typical dimensions (1mm)3 )
Consider nucleus of hydrogen in
H2O molecules:
proton magnetization randomly aligned
Macroscopic picture
(typical dimensions (1mm)3 )
Apply static magnetic field:
proton magnetization either aligns with
or against magnetic field
Bo
M
Macroscopic picture
(typical dimensions (1mm)3 )
Can perturb equilibrium by exciting at
Larmor frequency
w = (g /2 p) Bo
Can perturb equilibrium by exciting at
Larmor frequency
w = (g /2 p) Bo
Bo
Mxy
With correct strength and duration rf excitation
can flip magnetization
e.g. into the transverse plane
Spatial localization - reduce 3D to 2D
z
z
Bo
x
B
y
Spatial localization - reduce 3D to 2D
z
z
Bo
x
B
y
rf
Spatial localization - reduce 3D to 2D
z
z
Bo
x
B
y
Spatial localization - reduce 3D to 2D
z
z
Bo+Gz.z
x
B
y
Spatial localization - reduce 3D to 2D
z
z
resonance
condition
rf
Bo+Gz.z
x
B
y
Spatial localization - reduce 3D to 2D
y
z
z
y
Bo+Gz.z
x
B
x
MR pulse sequence
z
rf
Gx
Bo+ Gz.z
B
Gz
Gy
time
Spatial localization - e.g., in 1d what is r(x) ?
Once magnetization is in the transverse plane
it precesses at the Larmor frequency w = 2 p/g B(x)
M(x,t) = Mo r(x) exp(-i.g. f(x,t))
If we apply a linear gradient, Gx ,of magnetic field
along x the accumulated phase at x after time t will be:
t
f(x,t) = ∫o x Gx(t') dt'
(ignoring carrier term)
f
Spatial localization - What is r(x) ?
S(t)
object
B
x

no spatial information
Bo
xx
Spatial localization - What is r(x) ?
object
xx
B
Bo+Gxx
xx
Spatial localization - What is r(x) ?
S(t)
object
xx
B
Bo+Gxx
xx

Spatial localization - What is r(x) ?
S(t)
object
xx
B
Fourier
transform
Bo+Gxx
image
r(x)
xx
x
For an antenna sensitive to all the precessing
magnetization, the measured signal is:
S(t) = ∫ M(x,t) dx
= Mo ∫ r(x) exp (-i.(g. Gx) x.t) dx
therefore:
r(x) = ∫ M(x,t) dx
= Mo ∫ S(t) exp (i. c. x.t) dt
MR pulse sequence
rf
Gz
Gx
Gy
time
For NMR in a magnet with imperfect
homogeneity, spin coherence is lost because
of spatially varying precession
Hahn (UC Berkeley)showed that this could be
reversed by flipping the spins through 180° the spin echo
In MRI, spatially varying fields are applied
to provide spatial localization - these spatially
varying magnetic fields must also be
compensated - the gradient echo
MR pulse sequence
(centered echo)
rf
Gz
Gx
Gy
ADC
time
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
Gx
spins aligned
following excitation
Gx
dephasing
Gx
ADC
dephasing
Gx
ADC
rephasing
Gx
ADC
echo
rephased
Gx
ADC
Gx
ADC
+
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+ +
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+
ADC
+
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+
ADC
+
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+
+
+ +
+
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+
ADC
+
+
+
+
+ +
+
+
+
+
ADC
+
+
+
+
+ +
+
+
+
+
ADC
+
+
+
+
+ +
+
+
+
+
ADC
+
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+
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+
ADC
+
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ADC
+
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ADC
FOV
+
+
+
+
+ +
+
+
+
+
+
ADC
FOV
= resolution
N
FOV smaller than object
FOV
FOV
FOV smaller than object:
- wrap-around artifact
FOV
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
phase encoding 128
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
phase encoding 64
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
phase encoding 0
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
phase encoding -64
MR pulse sequence
for 2D
rf
Gz
Gx
Gy
ADC
time
phase encoding -127
k-space
Fourier
Fourier
Fourier transform(ed)
inner k-space
Fourier transform
overall contrast information
outer k-space
Fourier transform
edge information